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.S6 { margin: 2px 10px 9px 4px; padding: 0px; line-height: 21px; min-height: 0px; white-space: pre-wrap; color: rgb(0, 0, 0); font-family: Helvetica, Arial, sans-serif; font-style: normal; font-size: 14px; font-weight: 400; text-align: left;  }
.S7 { margin: 10px 10px 9px 4px; padding: 0px; line-height: 21px; min-height: 0px; white-space: pre-wrap; color: rgb(0, 0, 0); font-family: Helvetica, Arial, sans-serif; font-style: normal; font-size: 14px; font-weight: 400; text-align: left;  }</style></head><body><div class = rtcContent><h1  class = 'S0' id = 'T_2040B2DB' ><span>WBR系统建模分析</span></h1><div  class = 'S1'><div  class = 'S2'><span style=' font-weight: bold;'>目录</span></div><div  class = 'S3'><a href = "#H_C93962D5"><span>1 系统建模
</span></a><span>    </span><a href = "#H_5893A61A"><span>1.1 简化与假设
</span></a><span>    </span><a href = "#H_BA289B72"><span>1.2 变量、参数定义
</span></a><a href = "#H_E311880B"><span>2 运动学
</span></a><a href = "#H_3A53DB9A"><span>3 动力学
</span></a><span>    </span><a href = "#H_A46BB375"><span>3.1 动力学方程
</span></a><span>    </span><a href = "#H_A149B6C3"><span>3.2 方程求解
</span></a><a href = "#H_955EBF93"><span>4 状态空间模型</span></a></div></div><h2  class = 'S4' id = 'H_C93962D5' ><span>1 系统建模</span></h2><h3  class = 'S5' id = 'H_5893A61A' ><span>1.1 简化与假设</span></h3><div  class = 'S6'><span>(1) 机器人简化为机体、左右腿、左右驱动轮5部分组成的刚体系统</span></div><div  class = 'S6'><span>(2) 忽略腿部连杆机构运动（包括腿长变化，惯量变化）产生的动力学效应，即任意时刻均将腿部视为该时刻腿长对应参数的刚体</span></div><div  class = 'S6'><span>(3) 忽略腿部、机体位形变化对z轴转动惯量的影响</span></div><div  class = 'S6'><span>(4) 假设驱动轮与地面间无滑动（后续会加入离地/打滑检测）</span></div><div  class = 'S6'><span>(5) 滚转角roll（</span><span style="font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">ψ</span><span>）通过腿部控制器进行控制，在机器人整体动力学分析中假设 </span><span texencoding="\psi=\dot\psi=\ddot\psi=0" style="vertical-align:-5px"><img src="" width="96.5" height="19.5" /></span></div><div  class = 'S6'><span>(6) 腿部支持力通过腿部控制器进行控制，在机器人整体动力学分析中假设左右腿支持力保持相等，忽略侧向惯性力矩的影响。</span></div><h3  class = 'S5' id = 'H_BA289B72' ><span>1.2 变量、参数定义</span></h3><div  class = 'S6'><img class = "imageNode" src = "" width = "691" height = "137" alt = "model.jpg" style = "vertical-align: baseline; width: 691px; height: 137px;"></img></div><div  class = 'S6' id = 'H_58AC9401' ><span style=' font-weight: bold;'>变量（标有*号的为独立变量）</span></div><div  class = 'S6'><span>*</span><span texencoding="\theta_{w,i}\ (i=l,r)" style="vertical-align:-6px"><img src="" width="80.5" height="19.5" /></span><span>: 驱动轮转角（自然坐标系法向）</span></div><div  class = 'S6'><span>*</span><span texencoding="\theta_{l,i}\ (i=l,r)" style="vertical-align:-6px"><img src="" width="76.5" height="19.5" /></span><span>: 腿倾斜角（自然坐标系法向）</span></div><div  class = 'S6'><span>*</span><span texencoding="\theta_b" style="vertical-align:-6px"><img src="" width="15" height="19.5" /></span><span>: 机体倾斜角（自然坐标系法向）</span></div><div  class = 'S6'><span style="font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">ϕ</span><span>: 偏航角，yaw</span></div><div  class = 'S6'><span style="font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">s</span><span>: 自然坐标系下机器人水平方向移动距离</span></div><div  class = 'S6'><span texencoding="s_b" style="vertical-align:-6px"><img src="" width="14" height="19.5" /></span><span>: 自然坐标系下机体质心水平方向移动距离</span></div><div  class = 'S6'><span texencoding="h_b" style="vertical-align:-6px"><img src="" width="15.5" height="19.5" /></span><span>: 自然坐标系下机体质心竖直方向移动距离</span></div><div  class = 'S6'><span texencoding="s_{l,i}\ (i=l,r)" style="vertical-align:-6px"><img src="" width="75.5" height="19.5" /></span><span>: 自然坐标系下腿部质心水平方向移动距离</span></div><div  class = 'S6'><span texencoding="h__{l,i}\ (i=l,r)" style="vertical-align:-6px"><img src="" width="77" height="18.5" /></span><span>: 自然坐标系下腿部质心竖直方向移动距离</span></div><div  class = 'S6'><span>*</span><span texencoding="T__{w,i}\ (i=l,r)" style="vertical-align:-6px"><img src="" width="82" height="18.5" /></span><span>: 驱动轮转矩（腿-轮）</span></div><div  class = 'S6'><span>*</span><span texencoding="T__{b,i}\ (i=l,r)" style="vertical-align:-6px"><img src="" width="80" height="18.5" /></span><span>: 腿部转矩（机体-腿）</span></div><div  class = 'S6'><span texencoding="f_i\ (i=l,r)" style="vertical-align:-6px"><img src="" width="68.5" height="19.5" /></span><span>: 地面对驱动轮摩擦力</span></div><div  class = 'S6'><span texencoding="F__{ws,i}\ (i=l,r)" style="vertical-align:-6px"><img src="" width="86.5" height="18.5" /></span><span>: 驱动轮对腿水平方向作用力</span></div><div  class = 'S6'><span texencoding="F__{wh,i}\ (i=l,r)" style="vertical-align:-6px"><img src="" width="87.5" height="18.5" /></span><span>: 驱动轮对腿竖直方向作用力</span></div><div  class = 'S6'><span texencoding="F__{bs,i}\ (i=l,r)" style="vertical-align:-6px"><img src="" width="85" height="18.5" /></span><span>: 腿对机体水平方向作用力</span></div><div  class = 'S6'><span texencoding="F__{bh,i}\ (i=l,r)" style="vertical-align:-6px"><img src="" width="86" height="18.5" /></span><span>: 腿对机体竖直方向作用力</span></div><div  class = 'S6'><span style=' font-weight: bold;'>参数</span></div><div  class = 'S6'><span texencoding="R_w" style="vertical-align:-6px"><img src="" width="19" height="19.5" /></span><span>: 驱动轮半径</span></div><div  class = 'S6'><span texencoding="R_l" style="vertical-align:-6px"><img src="" width="15" height="19.5" /></span><span>: 驱动轮轮距/2</span></div><div  class = 'S6'><span texencoding="l_i\ (i=l,r)" style="vertical-align:-6px"><img src="" width="68.5" height="19.5" /></span><span>: 腿长*1</span></div><div  class = 'S6'><span texencoding="l_{w,i}\ (i=l,r)" style="vertical-align:-6px"><img src="" width="77.5" height="19.5" /></span><span>: 驱动轮到腿部质心距离*1</span></div><div  class = 'S6'><span texencoding="l_{b,i}\ (i=l,r)" style="vertical-align:-6px"><img src="" width="76" height="19.5" /></span><span>: 驱动轮到腿部质心距离*1</span></div><div  class = 'S6'><span texencoding="l_c" style="vertical-align:-6px"><img src="" width="11.5" height="19.5" /></span><span>: 机体质心到腿部关节距离</span></div><div  class = 'S6'><span texencoding="m_w" style="vertical-align:-6px"><img src="" width="20.5" height="19.5" /></span><span>: 驱动轮质量</span></div><div  class = 'S6'><span texencoding="m_l" style="vertical-align:-6px"><img src="" width="16.5" height="19.5" /></span><span>: 腿部质量</span></div><div  class = 'S6'><span texencoding="m_b" style="vertical-align:-6px"><img src="" width="19" height="19.5" /></span><span>: 机体质量</span></div><div  class = 'S6'><span texencoding="I_w" style="vertical-align:-6px"><img src="" width="14.5" height="19.5" /></span><span>: 驱动轮转动惯量（自然坐标系法向）</span></div><div  class = 'S6'><span texencoding="I_{l,i}" style="vertical-align:-6px"><img src="" width="16" height="19.5" /></span><span>: 腿部转动惯量（自然坐标系法向）*1</span></div><div  class = 'S6'><span texencoding="I_b" style="vertical-align:-6px"><img src="" width="13" height="19.5" /></span><span>: 机体转动惯量（自然坐标系法向）</span></div><div  class = 'S6'><span texencoding="I_z" style="vertical-align:-6px"><img src="" width="12" height="19.5" /></span><span>: 机器人z轴转动惯量*2</span></div><div  class = 'S6'><span style=' font-weight: bold;'>备注</span></div><div  class = 'S6'><span>*1：实际机器人腿部为五连杆机构，腿长可变，长度由腿部控制器控制，且质心位置、转动惯量均随腿长变化。</span></div><div  class = 'S6'><span>*2：绕z轴的转动惯量简化为常量。</span></div><h2  class = 'S4' id = 'H_E311880B' ><span>2 运动学</span></h2><div  class = 'S6'><span>根据变量、参数定义和几何关系得到系统的运动学方程，进而解得各变量的表达式</span></div><div  class = 'S6'><span texencoding="s=\frac{R_w}{2}(\theta_{w,l}+\theta_{w,r})" style="vertical-align:-15px"><img src="" width="114.5" height="37" /></span><span>    (2.1)</span></div><div  class = 'S6'><span texencoding="h_b=\frac{l_l}{2}cos\theta_{l,l}+\frac{l_r}{2}cos\theta_{l,r}" style="vertical-align:-15px"><img src="" width="144.5" height="37" /></span><span>    (2.2)</span></div><div  class = 'S6'><span texencoding="s_{l,i}=R_w\theta_{w,i}+l_{w,i}sin\theta_{l,i}" style="vertical-align:-6px"><img src="" width="136.5" height="19.5" /></span><span>    (2.3)</span></div><div  class = 'S6'><span texencoding="h_{l,i}=h_b-l_{b,i}cos\theta_{l,i}" style="vertical-align:-6px"><img src="" width="116.5" height="19.5" /></span><span>    (2.4)</span></div><div  class = 'S6'><span texencoding="R_w\theta_{w,l}=s_b-R_l\phi-l_lsin\theta_{l,l}" style="vertical-align:-6px"><img src="" width="163" height="19.5" /></span><span>    (2.5)</span></div><div  class = 'S6'><span texencoding="R_w\theta_{w,r}=s_b+R_l\phi-l_rsin\theta_{l,r}" style="vertical-align:-6px"><img src="" width="168" height="19.5" /></span><span>    (2.6)</span></div><div  class = 'S6'><span>联立式 (2.5)(2.6)</span></div><div  class = 'S6'><span texencoding="\phi=\frac{R_w}{2R_l}(-\theta_{w,l}+\theta_{w,r})-\frac{l_l}{2R_l}sin\theta_{l,l}+\frac{l_r}{2R_l}sin\theta_{l,r}" style="vertical-align:-17px"><img src="" width="277.5" height="39" /></span><span>    (2.7)</span></div><div  class = 'S6'><span texencoding="s_b=\frac{R_w}{2}(\theta_{w,l}+\theta_{w,r})+\frac{l_l}{2}sin\theta_{l,l}+\frac{l_r}{2}sin\theta_{l,r}" style="vertical-align:-15px"><img src="" width="241.5" height="37" /></span><span>    (2.8)</span></div><div  class = 'S6'><span>式 (2.1)(2.2)(2.3)(2.4)(2.7)(2.8) 对时间分别求一/二阶导</span></div><div  class = 'S6'><span texencoding="\dot{s}=\frac{R_w}{2}(\dot\theta_{w,l}+\dot\theta_{w,r})" style="vertical-align:-15px"><img src="" width="116" height="37" /></span><span>    (2.9)</span></div><div  class = 'S6'><span texencoding="\ddot{s}=\frac{R_w}{2}(\ddot\theta_{w,l}+\ddot\theta_{w,r})" style="vertical-align:-15px"><img src="" width="116" height="37" /></span><span>    (2.10)</span></div><div  class = 'S6'><span texencoding="\dot\phi=\frac{R_w}{2R_l}(-\dot\theta_{w,l}+\dot\theta_{w,r})-\frac{l_l}{2R_l}cos\theta_{l,l}\dot\theta_{l,l}+\frac{l_r}{2R_l}cos\theta_{l,r}\dot\theta_{l,r}" style="vertical-align:-17px"><img src="" width="317.5" height="39" /></span><span>    (2.11)</span></div><div  class = 'S6'><span texencoding="\ddot\phi=\frac{R_w}{2R_l}(-\ddot\theta_{w,l}+\ddot\theta_{w,r})-\frac{l_l}{2R_l}cos\theta_{l,l}\ddot\theta_{l,l}+\frac{l_r}{2R_l}cos\theta_{l,r}\ddot\theta_{l,r}+\frac{l_l}{2R_l}sin\theta_{l,l}\dot\theta_{l,l}^2-\frac{l_r}{2R_l}sin\theta_{l,r}\dot\theta_{l,r}^2" style="vertical-align:-17px"><img src="" width="496.5" height="39" /></span><span>    (2.12)</span></div><div  class = 'S6'><span texencoding="\dot{s}_b=\frac{R_w}{2}(\dot\theta_{w,l}+\dot\theta_{w,r})+\frac{l_l}{2}cos\theta_{l,l}\dot\theta_{l,l}+\frac{l_r}{2}cos\theta_{l,r}\dot\theta_{l,r}" style="vertical-align:-15px"><img src="" width="281.5" height="37" /></span><span>    (2.13)</span></div><div  class = 'S6'><span texencoding="\ddot{s}_b=\frac{R_w}{2}(\ddot\theta_{w,l}+\ddot\theta_{w,r})+\frac{l_l}{2}cos\theta_{l,l}\ddot\theta_{l,l}+\frac{l_r}{2}cos\theta_{l,r}\ddot\theta_{l,r}-\frac{l_l}{2}sin\theta_{l,l}\dot\theta_{l,l}^2-\frac{l_r}{2}sin\theta_{l,r}\dot\theta_{l,r}^2" style="vertical-align:-15px"><img src="" width="437.5" height="37" /></span><span>    (2.14)</span></div><div  class = 'S6'><span texencoding="\dot{h}_b=-\frac{l_l}{2}sin\theta_{l,l}\dot\theta_{l,l}-\frac{l_r}{2}sin\theta_{l,r}\dot\theta_{l,r}" style="vertical-align:-15px"><img src="" width="185.5" height="37" /></span><span>    (2.15)</span></div><div  class = 'S6'><span texencoding="\ddot{h}_b=-\frac{l_l}{2}sin\theta_{l,l}\ddot\theta_{l,l}-\frac{l_r}{2}sin\theta_{l,r}\ddot\theta_{l,r}-\frac{l_l}{2}cos\theta_{l,l}\dot\theta_{l,l}^2-\frac{l_r}{2}cos\theta_{l,r}\dot\theta_{l,r}^2" style="vertical-align:-15px"><img src="" width="346.5" height="37" /></span><span>    (2.16)</span></div><div  class = 'S6'><span texencoding="\dot{s}_{l,i}=R_w\dot\theta_{w,i}+l_{w,i}cos\theta_{l,i}\dot\theta_{l,i}" style="vertical-align:-6px"><img src="" width="155.5" height="23.5" /></span><span>    (2.17)</span></div><div  class = 'S6'><span texencoding="\ddot{s}_{l,i}=R_w\ddot\theta_{w,i}+l_{w,i}cos\theta_{l,i}\ddot\theta_{l,i}-l_{w,i}sin\theta_{l,i}\dot\theta_{l,i}^2" style="vertical-align:-7px"><img src="" width="237.5" height="24.5" /></span><span>    (2.18)</span></div><div  class = 'S6'><span texencoding="\dot{h}_{l,i}=\dot{h}_b+l_{b,i}sin\theta_{l,i}\dot\theta_{l,i}" style="vertical-align:-6px"><img src="" width="130" height="23.5" /></span><span>    (2.19)</span></div><div  class = 'S6'><span texencoding="\ddot{h}_{l,i}=\ddot{h}_b+l_{b,i}sin\theta_{l,i}\ddot\theta_{l,i}+l_{b,i}cos\theta_{l,i}\dot\theta_{l,i}^2" style="vertical-align:-7px"><img src="" width="212.5" height="24.5" /></span><span>    (2.20)</span></div><h2  class = 'S4' id = 'H_3A53DB9A' ><span>3 动力学</span></h2><h3  class = 'S5' id = 'H_A46BB375' ><span>3.1 动力学方程</span></h3><div  class = 'S6'><span>使用牛顿欧拉法建立系统的动力学方程</span></div><div  class = 'S6'><span>对于左右驱动轮 </span><span texencoding="(i=l,r)" style="vertical-align:-5px"><img src="" width="54.5" height="17.5" /></span></div><div  class = 'S6'><span texencoding="m_wR_w\ddot\theta_{w,i}=f_i-F_{ws,i}" style="vertical-align:-6px"><img src="" width="123.5" height="23.5" /></span><span>    (3.1)</span></div><div  class = 'S6'><span texencoding="I_w\ddot\theta_{w,i}=T_{lw,i}-f_iR_w" style="vertical-align:-6px"><img src="" width="116" height="23.5" /></span><span>    (3.2)</span></div><div  class = 'S6'><span>对于左右腿 </span><span texencoding="(i=l,r)" style="vertical-align:-5px"><img src="" width="54.5" height="17.5" /></span></div><div  class = 'S6'><span texencoding="m_l\ddot{s}_{l,i}=F_{ws,i}-F_{bs,i}" style="vertical-align:-6px"><img src="" width="114" height="21.5" /></span><span>    (3.3)</span></div><div  class = 'S6'><span texencoding="m_l\ddot{h}_{l,i}=F_{wh,i}-F_{bh,i}-m_lg" style="vertical-align:-6px"><img src="" width="156.5" height="23.5" /></span><span>    (3.4)</span></div><div  class = 'S6'><span texencoding="I_{l,i}\ddot\theta_{l,i}=(F_{wh,i}l_{w,i}+F_{bh,i}l_{b,i})sin\theta_{l,i}-(F_{ws,i}l_{w,i}+F_{bs,i}l_{b,i})cos\theta_{l,i}-T_{lw,i}+T_{bl,i}" style="vertical-align:-6px"><img src="" width="431" height="23.5" /></span><span>    (3.5)</span></div><div  class = 'S6'><span>对于机体</span></div><div  class = 'S6'><span texencoding="m_b\ddot{s}_b=F_{bs,l}+F_{bs,r}" style="vertical-align:-6px"><img src="" width="113.5" height="21.5" /></span><span>    (3.6)</span></div><div  class = 'S6'><span texencoding="m_b\ddot{h}_b=F_{bh,l}+F_{bh,r}-m_bg" style="vertical-align:-6px"><img src="" width="157.5" height="23.5" /></span><span>    (3.7)</span></div><div  class = 'S6'><span texencoding="I_b\ddot\theta_b=-(T_{bl,l}+T_{bl,r})-(F_{bs,l}+F_{bs,r})l_ccos\theta_b+(F_{bh,l}+F_{bh,r})l_csin\theta_b" style="vertical-align:-6px"><img src="" width="397" height="23.5" /></span><span>    (3.8)</span></div><div  class = 'S6'><span>机器人整体偏航（yaw）方向旋转</span></div><div  class = 'S6'><span texencoding="I_z\ddot\phi=(-f_l+f_r)R_l" style="vertical-align:-6px"><img src="" width="107" height="23.5" /></span><span>    (3.9)</span></div><div  class = 'S6'><span>左右腿支持力相等</span></div><div  class = 'S6'><span texencoding="F_{wh,l}=F_{wh,r}" style="vertical-align:-6px"><img src="" width="75.5" height="19.5" /></span><span>    （3.10）</span></div><h3  class = 'S5' id = 'H_A149B6C3' ><span>3.2 方程求解</span></h3><div  class = 'S6'><span>将式 (2.1)~(2.4)，(2.7)~(2.20) 带入式 (3.1)~(3.10) ，分析动力学方程组，共有15个未知量（5个广义坐标的二阶导数+10个约束力），15个独立方程（关于未知量线性），方程组有唯一解。</span></div><div  class = 'S7'><span>消去约束力，并对腿部、机体倾角进行小角度近似，得到如下方程组</span></div><div  class = 'S6'><span texencoding="(I_w\frac{l_l}{R_w}+m_wR_wl_l+m_lR_wl_{b,l})\ddot\theta__{w,l}+(m_ll_{w,l}l_{b,l}-I_{l,l})\ddot\theta_{l,l}+(m_ll_{w,l}+\frac{1}{2}m_bl_l)g\theta_{l,l}+T_{bl,l}-T_{lw,l}(1+\frac{l_l}{R_w})
=0" style="vertical-align:-17px"><img src="" width="589" height="39" /></span><span>    (3.11)</span></div><div  class = 'S6'><span texencoding="(I_w\frac{l_r}{R_w}+m_wR_wl_r+m_lR_wl_{b,r})\ddot\theta__{w,r}+(m_ll_{w,r}l_{b,r}-I_{l,r})\ddot\theta_{l,r}+(m_ll_{w,r}+\frac{1}{2}m_bl_r)g\theta_{l,r}+T_{bl,r}-T_{lw,r}(1+\frac{l_r}{R_w})
=0" style="vertical-align:-17px"><img src="" width="608" height="39" /></span><span>    (3.12)</span></div><div  class = 'S6'><span texencoding="-(m_wR_w^2+I_w+m_lR_w^2+\frac{1}{2}m_bR_w^2)\ddot\theta_{w,l}-(m_wR_w^2+I_w+m_lR_w^2+\frac{1}{2}m_bR_w^2)\ddot\theta_{w,r}-(m_lR_wl_{w,l}+\frac{1}{2}m_bR_wl_l)\ddot\theta_{l,l}-(m_lR_wl_{w,r}+\frac{1}{2}m_bR_wl_r)\ddot\theta_{l,r}+T_{lw,l}+T_{lw,r}
=0" style="vertical-align:-15px"><img src="" width="857.5" height="34" /></span><span>    (3.13)</span></div><div  class = 'S6'><span texencoding="(m_wR_wl_c+I_w\frac{l_c}{R_w}+m_lR_wl_c)\ddot\theta_{w,l}+(m_wR_wl_c+I_w\frac{l_c}{R_w}+m_lR_wl_c)\ddot\theta_{w,r}+m_ll_{w,l}l_c\ddot\theta_{l,l}+m_ll_{w,r}l_c\ddot\theta_{l,r}-I_b\ddot\theta_{l,r}+m_bgl_c\theta_b-(T_{lw,l}+T_{lw,r})\frac{l_c}{R_w}-(T_{bl,l}+T_{bl,r})
=0" style="vertical-align:-17px"><img src="" width="852" height="39" /></span><span>    (3.14)</span></div><div  class = 'S6'><span texencoding="(\frac{1}{2}I_z\frac{R_w}{R_l}+I_w\frac{R_l}{R_w})\ddot\theta_{w,l}-(\frac{1}{2}I_z\frac{R_w}{R_l}+I_w\frac{R_l}{R_w})\ddot\theta_{w,r}+\frac{1}{2}I_z\frac{l_l}{R_l}\ddot\theta_{l,l}-\frac{1}{2}I_z\frac{l_r}{R_l}\ddot\theta_{l,r}-T_{lw,l}\frac{R_l}{R_w}+T_{lw,r}\frac{R_l}{R_w}
=0" style="vertical-align:-17px"><img src="" width="537" height="39" /></span><span>    (3.15)</span></div><h2  class = 'S4' id = 'H_955EBF93' ><span>4 状态空间模型</span></h2><div  class = 'S6'><span>状态空间方程</span></div><div  class = 'S6'><span texencoding="\left\{\begin{array}{l} 
\mathbf{\dot{x}}=A\mathbf{x}+B\mathbf{u} \\ 
\mathbf{y}=C\mathbf{x}
\end{array}" style="vertical-align:-15px"><img src="" width="93.5" height="41" /></span><span>    (4.1)</span></div><div  class = 'S6'><span>式 (4.1) 中</span></div><div  class = 'S6'><span>状态向量 </span><span texencoding="\mathbf{x}=\left[\matrix{
s &amp; \dot{s} &amp; \phi &amp; \dot\phi &amp; \theta_{l,l} &amp; \dot\theta_{l,l} &amp; \theta_{l,r} &amp; \dot\theta_{l,r} &amp; \theta_b &amp; \dot\theta_b
}\right]^T" style="vertical-align:-7px"><img src="" width="279" height="28" /></span></div><div  class = 'S6'><span>控制向量 </span><span texencoding="\mathbf{u}=\left[\matrix{
T_{lw,l} &amp; T_{lw,r} &amp; T_{bl,l} &amp; T_{bl,r}
}\right]^T" style="vertical-align:-7px"><img src="" width="177.5" height="21" /></span></div><div  class = 'S6'><span>输出向量 </span><span texencoding="\mathbf{y}=\mathbf{x}" style="vertical-align:-5px"><img src="" width="36.5" height="17.5" /></span></div><div  class = 'S6'><span texencoding="A=\left[\matrix{
0&amp;1&amp;0&amp;0&amp;0&amp;0&amp;0&amp;0&amp;0&amp;0 \cr
0&amp;0&amp;0&amp;0&amp;a_{2,5}&amp;0&amp;a_{2,7}&amp;0&amp;a_{2,9}&amp;0 \cr
0&amp;0&amp;0&amp;1&amp;0&amp;0&amp;0&amp;0&amp;0&amp;0 \cr
0&amp;0&amp;0&amp;0&amp;a_{4,5}&amp;0&amp;a_{4,7}&amp;0&amp;a_{4,9}&amp;0 \cr
0&amp;0&amp;0&amp;0&amp;0&amp;1&amp;0&amp;0&amp;0&amp;0 \cr
0&amp;0&amp;0&amp;0&amp;a_{6,5}&amp;0&amp;a_{6,7}&amp;0&amp;a_{6,9}&amp;0 \cr
0&amp;0&amp;0&amp;0&amp;0&amp;0&amp;0&amp;1&amp;0&amp;0 \cr
0&amp;0&amp;0&amp;0&amp;a_{8,5}&amp;0&amp;a_{8,7}&amp;0&amp;a_{8,9}&amp;0 \cr
0&amp;0&amp;0&amp;0&amp;0&amp;0&amp;0&amp;0&amp;0&amp;1 \cr
0&amp;0&amp;0&amp;0&amp;a_{10,5}&amp;0&amp;a_{10,7}&amp;0&amp;a_{10,9}&amp;0 \cr
}\right]" style="vertical-align:-105px"><img src="" width="288" height="222" /></span><span>, </span><span texencoding="B=\left[\matrix{
0&amp;0&amp;0&amp;0 \cr
b_{2,1}&amp;b_{2,2}&amp;b_{2,3}&amp;b_{2,4} \cr
0&amp;0&amp;0&amp;0 \cr
b_{4,1}&amp;b_{4,2}&amp;b_{4,3}&amp;b_{4,4} \cr
0&amp;0&amp;0&amp;0 \cr
b_{6,1}&amp;b_{6,2}&amp;b_{6,3}&amp;b_{6,4} \cr
0&amp;0&amp;0&amp;0 \cr
b_{8,1}&amp;b_{8,2}&amp;b_{8,3}&amp;b_{8,4} \cr
0&amp;0&amp;0&amp;0 \cr
b_{10,1}&amp;b_{10,2}&amp;b_{10,3}&amp;b_{10,4} \cr
}\right]" style="vertical-align:-105px"><img src="" width="187.5" height="222" /></span><span>, </span><span texencoding="C =I_{10}" style="vertical-align:-6px"><img src="" width="48" height="19.5" /></span></div><div  class = 'S6'><span texencoding="a_{i,j}=\left\{\begin{array}{l}
\frac{R_w}{2}\left(\frac{\partial\ddot\theta_{w,l}}{\partial{x}_j}+\frac{\partial\ddot\theta_{w,r}}{\partial{x}_j}\right) &amp; (i=2) \\
\frac{R_w}{2R_l}\left(-\frac{\partial\ddot\theta_{w,l}}{\partial{x}_j}+\frac{\partial\ddot\theta_{w,r}}{\partial{x}_j}\right)-\frac{l_l}{2R_l}\frac{\partial\ddot\theta_{l,l}}{\partial{x}_j}+\frac{l_r}{2R_l}\frac{\partial\ddot\theta_{l,r}}{\partial{x}_j} &amp; (i=4) \\
\frac{\partial\dot{x}_i}{\partial{x}_j} &amp; (i=6,8,10)
\end{array}" style="vertical-align:-65px"><img src="" width="396.5" height="142" /></span></div><div  class = 'S6'><span texencoding="b_{i,j}=\left\{\begin{array}{l}
\frac{R_w}{2}\left(\frac{\partial\ddot\theta_{w,l}}{\partial{u}_j}+\frac{\partial\ddot\theta_{w,r}}{\partial{u}_j}\right) &amp; (i=2) \\
\frac{R_w}{2R_l}\left(-\frac{\partial\ddot\theta_{w,l}}{\partial{u}_j}+\frac{\partial\ddot\theta_{w,r}}{\partial{u}_j}\right)-\frac{l_l}{2R_l}\frac{\partial\ddot\theta_{l,l}}{\partial{u}_j}+\frac{l_r}{2R_l}\frac{\partial\ddot\theta_{l,r}}{\partial{u}_j} &amp; (i=4) \\
\frac{\partial\dot{x}_i}{\partial{u}_j} &amp; (i=6,8,10)
\end{array}" style="vertical-align:-65px"><img src="" width="396.5" height="142" /></span></div><div  class = 'S7'></div>
<br>
<!-- 
##### SOURCE BEGIN #####
%% WBR系统建模分析
%% 1 系统建模
% 1.1 简化与假设
% (1) 机器人简化为机体、左右腿、左右驱动轮5部分组成的刚体系统
% 
% (2) 忽略腿部连杆机构运动（包括腿长变化，惯量变化）产生的动力学效应，即任意时刻均将腿部视为该时刻腿长对应参数的刚体
% 
% (3) 忽略腿部、机体位形变化对z轴转动惯量的影响
% 
% (4) 假设驱动轮与地面间无滑动（后续会加入离地/打滑检测）
% 
% (5) 滚转角roll（$\psi$）通过腿部控制器进行控制，在机器人整体动力学分析中假设 $\psi=\dot\psi=\ddot\psi=0$
% 
% (6) 腿部支持力通过腿部控制器进行控制，在机器人整体动力学分析中假设左右腿支持力保持相等，忽略侧向惯性力矩的影响。
% 1.2 变量、参数定义
% 
% 
% *变量（标有*号的为独立变量）*
% 
% *$\theta_{w,i}\ (i=l,r)$: 驱动轮转角（自然坐标系法向）
% 
% *$\theta_{l,i}\ (i=l,r)$: 腿倾斜角（自然坐标系法向）
% 
% *$\theta_b$: 机体倾斜角（自然坐标系法向）
% 
% $\phi$: 偏航角，yaw
% 
% $s$: 自然坐标系下机器人水平方向移动距离
% 
% $s_b$: 自然坐标系下机体质心水平方向移动距离
% 
% $h_b$: 自然坐标系下机体质心竖直方向移动距离
% 
% $s_{l,i}\ (i=l,r)$: 自然坐标系下腿部质心水平方向移动距离
% 
% $h__{l,i}\ (i=l,r)$: 自然坐标系下腿部质心竖直方向移动距离
% 
% *$T__{w,i}\ (i=l,r)$: 驱动轮转矩（腿-轮）
% 
% *$T__{b,i}\ (i=l,r)$: 腿部转矩（机体-腿）
% 
% $f_i\ (i=l,r)$: 地面对驱动轮摩擦力
% 
% $F__{ws,i}\ (i=l,r)$: 驱动轮对腿水平方向作用力
% 
% $F__{wh,i}\ (i=l,r)$: 驱动轮对腿竖直方向作用力
% 
% $F__{bs,i}\ (i=l,r)$: 腿对机体水平方向作用力
% 
% $F__{bh,i}\ (i=l,r)$: 腿对机体竖直方向作用力
% 
% *参数*
% 
% $R_w$: 驱动轮半径
% 
% $R_l$: 驱动轮轮距/2
% 
% $l_i\ (i=l,r)$: 腿长*1
% 
% $l_{w,i}\ (i=l,r)$: 驱动轮到腿部质心距离*1
% 
% $l_{b,i}\ (i=l,r)$: 驱动轮到腿部质心距离*1
% 
% $l_c$: 机体质心到腿部关节距离
% 
% $m_w$: 驱动轮质量
% 
% $m_l$: 腿部质量
% 
% $m_b$: 机体质量
% 
% $I_w$: 驱动轮转动惯量（自然坐标系法向）
% 
% $I_{l,i}$: 腿部转动惯量（自然坐标系法向）*1
% 
% $I_b$: 机体转动惯量（自然坐标系法向）
% 
% $I_z$: 机器人z轴转动惯量*2
% 
% *备注*
% 
% *1：实际机器人腿部为五连杆机构，腿长可变，长度由腿部控制器控制，且质心位置、转动惯量均随腿长变化。
% 
% *2：绕z轴的转动惯量简化为常量。
%% 2 运动学
% 根据变量、参数定义和几何关系得到系统的运动学方程，进而解得各变量的表达式
% 
% $s=\frac{R_w}{2}(\theta_{w,l}+\theta_{w,r})$    (2.1)
% 
% $h_b=\frac{l_l}{2}cos\theta_{l,l}+\frac{l_r}{2}cos\theta_{l,r}$    (2.2)
% 
% $s_{l,i}=R_w\theta_{w,i}+l_{w,i}sin\theta_{l,i}$    (2.3)
% 
% $h_{l,i}=h_b-l_{b,i}cos\theta_{l,i}$    (2.4)
% 
% $R_w\theta_{w,l}=s_b-R_l\phi-l_lsin\theta_{l,l}$    (2.5)
% 
% $R_w\theta_{w,r}=s_b+R_l\phi-l_rsin\theta_{l,r}$    (2.6)
% 
% 联立式 (2.5)(2.6)
% 
% $\phi=\frac{R_w}{2R_l}(-\theta_{w,l}+\theta_{w,r})-\frac{l_l}{2R_l}sin\theta_{l,l}+\frac{l_r}{2R_l}sin\theta_{l,r}$    
% (2.7)
% 
% $s_b=\frac{R_w}{2}(\theta_{w,l}+\theta_{w,r})+\frac{l_l}{2}sin\theta_{l,l}+\frac{l_r}{2}sin\theta_{l,r}$    
% (2.8)
% 
% 式 (2.1)(2.2)(2.3)(2.4)(2.7)(2.8) 对时间分别求一/二阶导
% 
% $\dot{s}=\frac{R_w}{2}(\dot\theta_{w,l}+\dot\theta_{w,r})$    (2.9)
% 
% $\ddot{s}=\frac{R_w}{2}(\ddot\theta_{w,l}+\ddot\theta_{w,r})$    (2.10)
% 
% $\dot\phi=\frac{R_w}{2R_l}(-\dot\theta_{w,l}+\dot\theta_{w,r})-\frac{l_l}{2R_l}cos\theta_{l,l}\dot\theta_{l,l}+\frac{l_r}{2R_l}cos\theta_{l,r}\dot\theta_{l,r}$    
% (2.11)
% 
% $\ddot\phi=\frac{R_w}{2R_l}(-\ddot\theta_{w,l}+\ddot\theta_{w,r})-\frac{l_l}{2R_l}cos\theta_{l,l}\ddot\theta_{l,l}+\frac{l_r}{2R_l}cos\theta_{l,r}\ddot\theta_{l,r}+\frac{l_l}{2R_l}sin\theta_{l,l}\dot\theta_{l,l}^2-\frac{l_r}{2R_l}sin\theta_{l,r}\dot\theta_{l,r}^2$    
% (2.12)
% 
% $\dot{s}_b=\frac{R_w}{2}(\dot\theta_{w,l}+\dot\theta_{w,r})+\frac{l_l}{2}cos\theta_{l,l}\dot\theta_{l,l}+\frac{l_r}{2}cos\theta_{l,r}\dot\theta_{l,r}$    
% (2.13)
% 
% $\ddot{s}_b=\frac{R_w}{2}(\ddot\theta_{w,l}+\ddot\theta_{w,r})+\frac{l_l}{2}cos\theta_{l,l}\ddot\theta_{l,l}+\frac{l_r}{2}cos\theta_{l,r}\ddot\theta_{l,r}-\frac{l_l}{2}sin\theta_{l,l}\dot\theta_{l,l}^2-\frac{l_r}{2}sin\theta_{l,r}\dot\theta_{l,r}^2$    
% (2.14)
% 
% $\dot{h}_b=-\frac{l_l}{2}sin\theta_{l,l}\dot\theta_{l,l}-\frac{l_r}{2}sin\theta_{l,r}\dot\theta_{l,r}$    
% (2.15)
% 
% $\ddot{h}_b=-\frac{l_l}{2}sin\theta_{l,l}\ddot\theta_{l,l}-\frac{l_r}{2}sin\theta_{l,r}\ddot\theta_{l,r}-\frac{l_l}{2}cos\theta_{l,l}\dot\theta_{l,l}^2-\frac{l_r}{2}cos\theta_{l,r}\dot\theta_{l,r}^2$    
% (2.16)
% 
% $\dot{s}_{l,i}=R_w\dot\theta_{w,i}+l_{w,i}cos\theta_{l,i}\dot\theta_{l,i}$    
% (2.17)
% 
% $\ddot{s}_{l,i}=R_w\ddot\theta_{w,i}+l_{w,i}cos\theta_{l,i}\ddot\theta_{l,i}-l_{w,i}sin\theta_{l,i}\dot\theta_{l,i}^2$    
% (2.18)
% 
% $\dot{h}_{l,i}=\dot{h}_b+l_{b,i}sin\theta_{l,i}\dot\theta_{l,i}$    (2.19)
% 
% $\ddot{h}_{l,i}=\ddot{h}_b+l_{b,i}sin\theta_{l,i}\ddot\theta_{l,i}+l_{b,i}cos\theta_{l,i}\dot\theta_{l,i}^2$    
% (2.20)
%% 3 动力学
% 3.1 动力学方程
% 使用牛顿欧拉法建立系统的动力学方程
% 
% 对于左右驱动轮 $(i=l,r)$
% 
% $m_wR_w\ddot\theta_{w,i}=f_i-F_{ws,i}$    (3.1)
% 
% $I_w\ddot\theta_{w,i}=T_{lw,i}-f_iR_w$    (3.2)
% 
% 对于左右腿 $(i=l,r)$
% 
% $m_l\ddot{s}_{l,i}=F_{ws,i}-F_{bs,i}$    (3.3)
% 
% $m_l\ddot{h}_{l,i}=F_{wh,i}-F_{bh,i}-m_lg$    (3.4)
% 
% $I_{l,i}\ddot\theta_{l,i}=(F_{wh,i}l_{w,i}+F_{bh,i}l_{b,i})sin\theta_{l,i}-(F_{ws,i}l_{w,i}+F_{bs,i}l_{b,i})cos\theta_{l,i}-T_{lw,i}+T_{bl,i}$    
% (3.5)
% 
% 对于机体
% 
% $m_b\ddot{s}_b=F_{bs,l}+F_{bs,r}$    (3.6)
% 
% $m_b\ddot{h}_b=F_{bh,l}+F_{bh,r}-m_bg$    (3.7)
% 
% $I_b\ddot\theta_b=-(T_{bl,l}+T_{bl,r})-(F_{bs,l}+F_{bs,r})l_ccos\theta_b+(F_{bh,l}+F_{bh,r})l_csin\theta_b$    
% (3.8)
% 
% 机器人整体偏航（yaw）方向旋转
% 
% $I_z\ddot\phi=(-f_l+f_r)R_l$    (3.9)
% 
% 左右腿支持力相等
% 
% $F_{wh,l}=F_{wh,r}$    （3.10）
% 3.2 方程求解
% 将式 (2.1)~(2.4)，(2.7)~(2.20) 带入式 (3.1)~(3.10) ，分析动力学方程组，共有15个未知量（5个广义坐标的二阶导数+10个约束力），15个独立方程（关于未知量线性），方程组有唯一解。
%% 
% 消去约束力，并对腿部、机体倾角进行小角度近似，得到如下方程组
% 
% $(I_w\frac{l_l}{R_w}+m_wR_wl_l+m_lR_wl_{b,l})\ddot\theta__{w,l}+(m_ll_{w,l}l_{b,l}-I_{l,l})\ddot\theta_{l,l}+(m_ll_{w,l}+\frac{1}{2}m_bl_l)g\theta_{l,l}+T_{bl,l}-T_{lw,l}(1+\frac{l_l}{R_w})=0$    
% (3.11)
% 
% $(I_w\frac{l_r}{R_w}+m_wR_wl_r+m_lR_wl_{b,r})\ddot\theta__{w,r}+(m_ll_{w,r}l_{b,r}-I_{l,r})\ddot\theta_{l,r}+(m_ll_{w,r}+\frac{1}{2}m_bl_r)g\theta_{l,r}+T_{bl,r}-T_{lw,r}(1+\frac{l_r}{R_w})=0$    
% (3.12)
% 
% $-(m_wR_w^2+I_w+m_lR_w^2+\frac{1}{2}m_bR_w^2)\ddot\theta_{w,l}-(m_wR_w^2+I_w+m_lR_w^2+\frac{1}{2}m_bR_w^2)\ddot\theta_{w,r}-(m_lR_wl_{w,l}+\frac{1}{2}m_bR_wl_l)\ddot\theta_{l,l}-(m_lR_wl_{w,r}+\frac{1}{2}m_bR_wl_r)\ddot\theta_{l,r}+T_{lw,l}+T_{lw,r}=0$    
% (3.13)
% 
% $(m_wR_wl_c+I_w\frac{l_c}{R_w}+m_lR_wl_c)\ddot\theta_{w,l}+(m_wR_wl_c+I_w\frac{l_c}{R_w}+m_lR_wl_c)\ddot\theta_{w,r}+m_ll_{w,l}l_c\ddot\theta_{l,l}+m_ll_{w,r}l_c\ddot\theta_{l,r}-I_b\ddot\theta_{l,r}+m_bgl_c\theta_b-(T_{lw,l}+T_{lw,r})\frac{l_c}{R_w}-(T_{bl,l}+T_{bl,r})=0$    
% (3.14)
% 
% $(\frac{1}{2}I_z\frac{R_w}{R_l}+I_w\frac{R_l}{R_w})\ddot\theta_{w,l}-(\frac{1}{2}I_z\frac{R_w}{R_l}+I_w\frac{R_l}{R_w})\ddot\theta_{w,r}+\frac{1}{2}I_z\frac{l_l}{R_l}\ddot\theta_{l,l}-\frac{1}{2}I_z\frac{l_r}{R_l}\ddot\theta_{l,r}-T_{lw,l}\frac{R_l}{R_w}+T_{lw,r}\frac{R_l}{R_w}=0$    
% (3.15)
%% 4 状态空间模型
% 状态空间方程
% 
% $\left\{\begin{array}{l} \mathbf{\dot{x}}=A\mathbf{x}+B\mathbf{u} \\ \mathbf{y}=C\mathbf{x}\end{array}$    
% (4.1)
% 
% 式 (4.1) 中
% 
% 状态向量 $\mathbf{x}=\left[\matrix{s & \dot{s} & \phi & \dot\phi & \theta_{l,l} 
% & \dot\theta_{l,l} & \theta_{l,r} & \dot\theta_{l,r} & \theta_b & \dot\theta_b}\right]^T$
% 
% 控制向量 $\mathbf{u}=\left[\matrix{T_{lw,l} & T_{lw,r} & T_{bl,l} & T_{bl,r}}\right]^T$
% 
% 输出向量 $\mathbf{y}=\mathbf{x}$
% 
% $A=\left[\matrix{0&1&0&0&0&0&0&0&0&0 \cr0&0&0&0&a_{2,5}&0&a_{2,7}&0&a_{2,9}&0 
% \cr0&0&0&1&0&0&0&0&0&0 \cr0&0&0&0&a_{4,5}&0&a_{4,7}&0&a_{4,9}&0 \cr0&0&0&0&0&1&0&0&0&0 
% \cr0&0&0&0&a_{6,5}&0&a_{6,7}&0&a_{6,9}&0 \cr0&0&0&0&0&0&0&1&0&0 \cr0&0&0&0&a_{8,5}&0&a_{8,7}&0&a_{8,9}&0 
% \cr0&0&0&0&0&0&0&0&0&1 \cr0&0&0&0&a_{10,5}&0&a_{10,7}&0&a_{10,9}&0 \cr}\right]$, 
% $B=\left[\matrix{0&0&0&0 \crb_{2,1}&b_{2,2}&b_{2,3}&b_{2,4} \cr0&0&0&0 \crb_{4,1}&b_{4,2}&b_{4,3}&b_{4,4} 
% \cr0&0&0&0 \crb_{6,1}&b_{6,2}&b_{6,3}&b_{6,4} \cr0&0&0&0 \crb_{8,1}&b_{8,2}&b_{8,3}&b_{8,4} 
% \cr0&0&0&0 \crb_{10,1}&b_{10,2}&b_{10,3}&b_{10,4} \cr}\right]$, $C =I_{10}$
% 
% $$a_{i,j}=\left\{\begin{array}{l}\frac{R_w}{2}\left(\frac{\partial\ddot\theta_{w,l}}{\partial{x}_j}+\frac{\partial\ddot\theta_{w,r}}{\partial{x}_j}\right) 
% & (i=2) \\\frac{R_w}{2R_l}\left(-\frac{\partial\ddot\theta_{w,l}}{\partial{x}_j}+\frac{\partial\ddot\theta_{w,r}}{\partial{x}_j}\right)-\frac{l_l}{2R_l}\frac{\partial\ddot\theta_{l,l}}{\partial{x}_j}+\frac{l_r}{2R_l}\frac{\partial\ddot\theta_{l,r}}{\partial{x}_j} 
% & (i=4) \\\frac{\partial\dot{x}_i}{\partial{x}_j} & (i=6,8,10)\end{array}$$
% 
% $$b_{i,j}=\left\{\begin{array}{l}\frac{R_w}{2}\left(\frac{\partial\ddot\theta_{w,l}}{\partial{u}_j}+\frac{\partial\ddot\theta_{w,r}}{\partial{u}_j}\right) 
% & (i=2) \\\frac{R_w}{2R_l}\left(-\frac{\partial\ddot\theta_{w,l}}{\partial{u}_j}+\frac{\partial\ddot\theta_{w,r}}{\partial{u}_j}\right)-\frac{l_l}{2R_l}\frac{\partial\ddot\theta_{l,l}}{\partial{u}_j}+\frac{l_r}{2R_l}\frac{\partial\ddot\theta_{l,r}}{\partial{u}_j} 
% & (i=4) \\\frac{\partial\dot{x}_i}{\partial{u}_j} & (i=6,8,10)\end{array}$$
%% 
%
##### SOURCE END #####
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